cp's OEIS Frontend

A111138 Let b(n) denote the number of nontriangular numbers less than or equal to n. Then a(n) = b(n-1) + a(b(n-1)), with a(1) = a(2) = 0, a(3) = 1.

%I A111138 #31 Oct 27 2024 09:27:39
%S A111138 0,0,1,1,2,4,4,5,7,10,10,11,13,16,20,20,21,23,26,30,35,35,36,38,41,45,
%T A111138 50,56,56,57,59,62,66,71,77,84,84,85,87,90,94,99,105,112,120,120,121,
%U A111138 123,126,130,135,141,148,156,165,165,166,168,171,175,180,186,193,201
%N A111138 Let b(n) denote the number of nontriangular numbers less than or equal to n. Then a(n) = b(n-1) + a(b(n-1)), with a(1) = a(2) = 0, a(3) = 1.
%C A111138 For a subgroup H of order p^n (p an odd prime) of the subgroup generated by all commutators [x_j,x_i] in the relatively free group F of class three and exponent p, freely generated by x_1, x_2,..., x_k, (k sufficiently large) the minimum size of the subgroup of [H,F] of F_3 is p^{kn - a(n)}.
%C A111138 The sequence arises when finding a purely numerical sufficient condition for the capability of p-groups of class two and exponent p, where p is an odd prime.
%C A111138 Partial sums of A002262. - _Gionata Neri_, Sep 04 2015
%H A111138 Arturo Magidin, <a href="http://arxiv.org/abs/math.GR/0506578">Capable groups of prime exponent and class two II</a>, arXiv:math/0506578 [math.GR], 2005.
%F A111138 If we write n = (m choose 2) + s, 0<=s<=m, then a(n)=(m choose 3) + (s choose 2).
%F A111138 a(N) = Comb(T,2)+Comb(R,3) where R:=Round(Sqrt(2*N)) and T:=N-Comb(R,2). - _Gerald Hillier_, Nov 18 2017
%e A111138 a(31) = b(30) + a(b(30)) = 23 + a(23) = 23 + b(22) + a(b(22)) = 23 + 16 + a(16) = 39 + b(15) + a(b(15)) = 39 + 10 + a(10) = 49 + b(9) + a(b(9)) = 49 + 6 + a(6) = 55 + b(5) + a(b(5)) = 55 + 3 + a(3) = 58 + 1 = 59.
%t A111138 a[1] = a[2] = 0; a[3] = 1; a[n_] := a[n] = b[n - 1] + a[b[n - 1]]; b[n_] := n - Floor[(Sqrt[8n + 1] - 1)/2]; Array[a, 64] (* _Robert G. Wilson v_, Feb 01 2006 *)
%o A111138 (PARI) a(n) = my(r,m=sqrtint(n<<1,&r)); if(r<m, r+=m, r-=m;m++); binomial(m,3) + binomial(r>>1,2); \\ _Kevin Ryde_, Oct 26 2024
%Y A111138 Cf. A083920.
%K A111138 nonn,easy
%O A111138 1,5
%A A111138 _Arturo Magidin_, Oct 17 2005; definition corrected Feb 01 2006
%E A111138 More terms from _Robert G. Wilson v_, Feb 01 2006